Hyperbolic components of rational maps: Quantitative equidistribution and counting

Abstract

Let $\Lambda$ be a quasi-projective variety and assume that, either $\Lambda$ is a subvariety of the moduli space ${\mathcal{M}}_d$ of degree $d$ rational maps, or $\Lambda$ parametrizes an algebraic family $(f_{\lambda }{)}_{\lambda \in \Lambda }$ of degree $d$ rational maps on ${\mathbb{P}}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the ${\log }^{+}$ of the modulus of the multipliers of periodic points.

Typo corrected in the title. Comments are welcome!